A075507 Shifts one place left under 8th-order binomial transform.

1, 1, 9, 89, 1009, 13457, 210105, 3747753, 74565473, 1628999841, 38704241897, 993034281593, 27340167242321, 803154583649329, 25050853217628313, 826165199464341705, 28707262835597618369, 1047731789671001235265, 40053733152627299592137, 1599910554128824794493593

Offset: 0

Author

Wolfdieter Lang, Oct 02 2002

Keywords

• nonn
• easy
• eigen

Comments

Previous name was: Row sums of triangle A075503 (for n>=1).

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..110

Crossrefs

Shifts one place left under k-th order binomial transform, k=1..10: A000110, A004211, A004212, A004213, A005011, A005012, A075506, A075507, A075508, A075509.

Programs

GAP

List([0..20],n->Sum([0..n],m->8^(n-m)*Stirling2(n,m))); # Muniru A Asiru, Mar 20 2018

Maple

[seq(factorial(k)*coeftayl(exp((exp(8*x)-1)/8), x = 0, k), k=0..20)]; # Muniru A Asiru, Mar 20 2018

Mathematica

Table[8^n BellB[n, 1/8], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)

Formula

a(n) = Sum_{m=0..n} 8^(n-m)*S2(n,m), with S2(n,m) = A008277(n,m) (Stirling2).

E.g.f.: exp((exp(8*x)-1)/8).

O.g.f.: Sum_{k>=0} x^k/Product_{j=1..k} (1 - 8*j*x). - Ilya Gutkovskiy, Mar 20 2018

a(n) ~ 8^n * n^n * exp(n/LambertW(8*n) - 1/8 - n) / (sqrt(1 + LambertW(8*n)) * LambertW(8*n)^n). - Vaclav Kotesovec, Jul 15 2021

Extensions

a(0)=1 inserted and new name by Vladimir Reshetnikov, Oct 20 2015